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M O D E L S O F Q U A N T U M M AT T E R

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Models of Quantum Matter A First Course on Integrability and the Bethe Ansatz Hans-Peter Eckle Humboldt Study Centre Ulm University

1

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3

Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford.

It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Hans-Peter Eckle 2019 The moral rights of the author have been asserted First Edition published in 2019 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2019930263 ISBN 978–0–19–967883–9 DOI: 10.1093/oso/9780199678839.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Für Irene

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Preface

Why study Bethe ansatz? The Bethe ansatz provides one of the very few methodologies to calculate the physical properties of models for strongly interacting quantum matter non– perturbatively. Arguably it is the only such method we have which is exact. This means, once we have set up the model, there are no approximations or further assumptions necessary: we can exactly compute physically relevant properties of the model. There is, furthermore, an infinite set of conserved quantities: the quantum mechanical model is integrable.

This makes the search for quantum models which are amenable to an exact solution by the Bethe ansatz methodology so important and rewarding. Even if—as is sometimes, but certainly not always, the case—the model with an exact solution is not the most physically relevant one, the exact solution will provide important benchmarks for other models that may occasionally be closer to physical reality, but do not admit an exact solution. Hence, for a plethora of cases, the Bethe ansatz solution provides valuable insight into the physics of strongly interacting quantum matter.

Since Hans Bethe provided the eponymous method to solve the Heisenberg quantum spin chain, many more models of low-dimensional quantum systems have been found to be integrable by the Bethe ansatz.

Moreover, these models and their integrability have been and continue to play an influential role in many subfields of physics, which include classical and quantum statistical mechanics, quantum field theory, and quantum many-particle and condensed matter physics, the latter in recent times especially in connection with modern developments in physics on the nanometre scale and in low dimensions. Quantum optics has also benefited from studying integrable models, especially in investigations of ultracold Bosonic and Fermionic quantum gases and Bose–Einstein condensates in artificial crystals of light, the so-called optical lattices. Recently in string theory and cosmology there is a hype of activity involving conjectures of Bethe ansatz integrability in the framework of the celebrated anti-de-Sitter space/conformal field theory (AdS/CFT) correspondence.

Of course, Bethe ansatz and integrability are discussed in Mathematical Physics, but there is also an ongoing cross-fertilization with various subfields of pure Mathematics.

Some prominent examples of integrable models include: various variants of the Heisenberg quantum spin chain whose physical realizations are probed by neutron scattering; the Hubbard model and its variants which inter alia have been discussed in connection with high-temperature superconductivity; the Kondo model which has recently seen a renaissance because of the development of tunable quantum dots; interacting Bose and Fermi gases which can now be produced in very pure and tunable form in optical lattices.

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Preface

But, what is the Bethe ansatz? In its original form, devised by Bethe, it is an ingeniously guessed form for the wave function of a one-dimensional quantum system. However, why this wave function is correct and even exact remained an open question which has only been answered much later through the algebraic form of the Bethe ansatz. This method enables us to construct an integrable quantum model in one dimension from a two-dimensional statistical mechanical model. The construction reveals the reason for quantum integrability and delivers the infinite set of conserved quantities together with the wave function which Bethe guessed. In our exposition of the Bethe ansatz methodology, we shall therefore start, somewhat unhistorically, but more systematically, with the algebraic Bethe ansatz.

Who is this book for? Interestingly, a look at Richard Feynman’s last blackboards (Paz, 1989) reveals that he may well have been interested. In one of his last publications (1988), Feynman in fact wrote: ‘I got really fascinated by these (1 + 1)-dimensional models that are solved by the Bethe ansatz and how mysteriously they jump out at you and work and you don’t know why. I am trying to understand all this better.’ Cited after Batchelor (2007).

In view of the exciting developments in Bethe ansatz of the last two decades, Richard Feynman’s fascination would certainly have continued.

The most helpful prerequisites for present readers are a good grounding in quantum mechanics, statistical mechanics, and the basics of quantum many-particle theory, especially second quantization. However, we shall comprehensively discuss the necessary tools and background in part I of the book. Through this approach, the book should be smoothly accessible for Master’s students who look for an area of specialization as well as for beginning graduate students. Moreover, to paraphrase Paul Halmos (in the preface of his book on Measure Theory (Halmos, 1978)), the novice to the Bethe ansatz methodology should not be discouraged if she or he finds that she or he does not have the prerequisites to read the preliminaries. After all, as Max Born reminds us, where would quantum physics be if Werner Heisenberg had been discouraged that he did not know what a matrix was when he developed the matrix form of quantum mechanics?

The book grew out of lecture notes the author prepared for an invited graduate lecture series at the Indian Institute of Science in Bangalore in 1995, summer school courses at the University of Jyväskylä in Central Finland in 1997 (on Bethe Ansatz Methods in Many–Body Physics) and 1999 (on Conformal Invariance in Statistical Physics), a graduate course at the same University which, together with an amiable group of students, made the extremely cold Finnish winter of 1999 actually an enjoyable experience, and postgraduate courses at the University of New South Wales in Sydney in 2000, and within the Mathematics–Physics MP 2 Platform at Göteborg University in 2009, as well as summer school lectures in Turkey: 2013 in Turunç, 2014 in Izmir and 2013 in Ireland in Dungarven under the auspices of the School of Theoretical Physics of the Dublin Institute for Advanced Studies.

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Acknowledgements

The author’s working knowledge of the topics and methods on which this book focuses was shaped through collaborations with several people who are gratefully acknowledged in chronological order: Ferenc Woynarovich, Tuong Truong, Chris Hamer, Rudolf Römer, Bill Sutherland, Alexander Punnoose, Henrik Johannesson, Charles Stafford, Tim Byrnes, Robert Bursill, Anders Sandvik, and Johan Nilsson. Gratefully acknowledging discussions with many others, especially my thesis advisor Dieter Schotte and with Ingo Peschel and Karol Penson, I shall not attempt to mention all of them for fear that someone’s name might escape my attention.

I learned coordinate Bethe ansatz from and during a fruitful collaboration with Ferenc Woynarovich. We both were initiated to the algebraic Bethe ansatz by Tuong Truong.

I also often enjoyed stimulating discussions with Holger Frahm as well as mutual visits with Fabian Essler, Andreas Klümper, and Andreas Schadschneider.

My deeply felt thanks go to Henrik Johannesson for his kind invitations to prepare and deliver lectures in Göteborg and, very especially, for our long-standing collaboration and friendship.

Great thanks are also due to Henrik again and Alexander Stolin for organizing and co–lecturing the course on ‘Integrable Models and Quantum Groups’ within the Mathematics–Physics MP 2 Platform at Göteborg University.

Jussi Timonen’s initiative is highly appreciated for suggesting and encouraging my lectures in Jyväskylä.

Valuable discussions with Balazs Hetényi, especially in connection with the coorganization of the summer schools in Turkey, and with Daniel Braak, are gratefully acknowledged. Many thanks also go to Karsten Seifert who volunteered to undertake the altruistic task of proof-reading parts of this book at various stages of its formation.

Many thanks are due to Natan Andrei, Nikolay Bogoliubov, and the late Anatoli Izergin for their hospitality at Rutgers and in Sankt Petersburg, respectively, and the sharing of their knowledge in stimulating personal discussions and in the written form of lecture notes, and also to Vladimir Korepin for encouragement on many occasions.

Many thanks are also due to Peter Horsch, Joachim Stolze, and Johannes Voit, and the late Heinz Barentzen and Hellmut Keiter for discussions and hospitality in Stuttgart, Bayreuth, and Dortmund, respectively, and the sharing of pertinent sets of lecture notes.

The team at Oxford University Press cannot be praised highly enough, especially their friendly patience and cheerful spirit, which appear to be inexhaustible. The incentive of April Warman got this project under way and helped it along in its early infant steps.

Sönke Adlung supplied fresh guidance to help the project along whenever there seemed to be an impasse. Ania Wronksi never tired of providing motivation and momentum for the book project and, of course, all the necessary information.

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Contents

1 Introduction

1

Part 1 Methods and Models in the Theory of Quantum Matter 2 Quantum Many-Particle Systems and Second Quantization 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

Many-particle Hilbert spaces Occupation number representation: Bosons and Fermions Creation and annihilation operators for Bosons Basis transformation Quantum field operators One-particle operators Two-particle operators Second quantization of the Schrödinger equation: Bosonic case Creation and annihilation operators for Fermions Second quantization of the Schrödinger equation: Fermionic case Second quantization formalism and the many-particle wave function Normal ordering

3 Angular Momentum 3.1 Angular momentum of a single quantum particle 3.2 Angular momentum of several quantum particles

4 Equilibrium Statistical Mechanics 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13

Fundamental postulate of equilibrium statistical mechanics Microcanonical ensemble Entropy Second law of thermodynamics Temperature Pressure and chemical potential First law of thermodynamics Canonical ensemble Partition function Grand canonical ensemble Gibbs entropy Density matrix Non-interacting quantum gases

5 8 19 21 26 27 29 33 35 36 40 41 43

45 45 56

63 64 66 67 68 69 69 70 70 72 75 77 78 82

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Contents

4.14 4.15 4.16 4.17

Classical lattice models in statistical mechanics Interacting magnetic moments: mean-field theory Transfer matrix Exact solution of the one-dimensional Ising model

5 Phase Transitions, Critical Phenomena, and Finite-Size Scaling 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Phases, phase diagrams, and phase transition Critical behaviour Landau–Ginzburg theory Scaling Renormalization group Finite-size scaling Quantum phase transitions

6 Statistical Mechanics and Quantum Field Theory 6.1 Connection between statistical mechanics and quantum field theory 6.2 Thermal fluctuations and quantum fluctuations

7 Conformal Symmetry in Statistical Mechanics 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

From scale to conformal invariance Conformal symmetry Conformal transformations in dimensions larger than two Conformal transformations in two dimensions Order parameter fields and correlation functions Energy-momentum tensor Energy-momentum tensor in two dimensions Conformal Ward identities Energy-momentum tensor and the Virasoro algebra Finite-size corrections revisited

8 Models of Strongly Interacting Quantum Matter 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

Bose fluid Electronic correlations Coulomb gas Landau Fermi liquid theory Luttinger liquid theory Magnetism Hubbard model Heisenberg model Magnetic quantum impurity models Quantum Rabi model

93 97 106 108

111 113 116 120 126 131 147 154

177 177 181

188 189 194 195 202 205 209 212 213 218 234

241 242 255 269 278 289 313 356 366 372 395

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xiii

Part 2 Algebraic Bethe Ansatz 9 Ice Model 9.1 Physical motivation for the square lattice ice model 9.2 Definition of the ice model

10 General Square Lattice Vertex Models 10.1 Vertex models in two dimensions 10.2 Sixteen- and eight-vertex models 10.3 Vertex Boltzmann weights and the partition function 10.4 R-matrix: matrix of Boltzmann weights of a vertex 10.5 Integrability and the transfer matrix 10.6 Commuting transfer matrices 10.7 Monodromy matrix 10.8 Further to the L-operator 10.9 Yang–Baxter relations 10.10 More on Yang–Baxter relations 10.11 Exploiting Yang–Baxter integrability

11 Six-Vertex Model 11.1 11.2 11.3 11.4 11.5

Yang–Baxter relation for the six-vertex model Parameterization of the six-vertex model Algebraic Bethe ansatz solution of the six-vertex model Quantum Hamiltonians from the transfer matrix Inhomogeneous Yang–Baxter quantum integrable models

12 Quantum Tavis–Cummings Model 12.1 12.2 12.3 12.4 12.5 12.6

Algebraic Bethe ansatz revisited Modified quantum Tavis–Cummings model Transfer matrix of the modified quantum Tavis–Cummings model Commutativity of the transfer matrix Simple quantum Tavis–Cummings model Bethe ansatz solution of the quantum Tavis–Cummings model

425 425 428

430 432 432 432 434 438 438 440 446 447 449 451

454 455 456 459 467 470

474 474 475 476 479 481 484

Part 3 Coordinate Bethe Ansatz 13 The Anisotropic Heisenberg Quantum Spin Chain 13.1 Description of the XXZ Heisenberg quantum spin chain 13.2 Special cases of the XXZ Heisenberg quantum spin chain 13.3 Basic properties of the XXZ Heisenberg quantum spin chain

14 Bethe Ansatz for the Anisotropic Heisenberg Quantum Spin Chain 14.1 Verification of the Bethe ansatz 14.2 Periodic boundary conditions 14.3 Parameterization of the quasi-momenta

491 492 495 498

502 503 508 511

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14.4 Ground state of the XXZ Heisenberg quantum spin chain 14.5 Excitations of the XXZ Heisenberg quantum spin chain 14.6 Excitations of the isotropic antiferromagnetic spin chain

15 Bose Gas in One Dimension: Lieb–Liniger Model 15.1 Classical non-linear Schrödinger equation 15.2 Quantum non-linear Schrödinger model 15.3 Lieb–Liniger model in the hard core limit: gas 15.4 δ-potential as boundary condition 15.5 Form of the Bethe eigenfunctions 15.6 Construction of the Bethe ansatz wave function 15.7 Unrestricted configuration space 15.8 Periodic boundary conditions: Bethe ansatz equations 15.9 Ground state of the δ-Bose gas in the thermodynamic limit 15.10 Excited states

517 522 537

545 546 547 549 552 553 558 562 565 568 573

Part 4 Electronic Systems: Nested Bethe Ansatz 16 Electronic Systems 16.1 16.2 16.3 16.4 16.5

Fermi gas in one dimension One-dimensional Hubbard model Kondo model Algebraic Bethe ansatz for the spin eigenvalue problem Magnetic impurities in nanostructures: Bethe ansatz results

585 586 598 600 612 625

Part 5 Thermodynamic Bethe Ansatz 17 Thermodynamics of the Repulsive Lieb–Liniger Model 17.1 Thermodynamic limit, particles, and holes 17.2 Bethe ansatz equations for particles and holes 17.3 Entropy and thermodynamic Bethe ansatz

18 Thermodynamics of the Isotropic Heisenberg Quantum Spin Chain 18.1 18.2 18.3 18.4 18.5

Summary of Bethe ansatz for the isotropic spin chain Preparation of the thermodynamic Bethe ansatz: particles and holes Thermodynamic Bethe ansatz equations Thermodynamics Thermodynamics for small T and h

633 633 634 636

641 641 644 649 652 653

Part 6 Bethe Ansatz for Finite Systems 19 Mathematical Tools 19.1 Euler–Maclaurin formula 19.2 Wiener–Hopf technique

657 657 661

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20 Finite Heisenberg Quantum Spin Chain 20.1 20.2 20.3 20.4 20.5

Bethe ansatz for the finite Heisenberg quantum spin chain Finite-size corrections Application of the Euler–Maclaurin formula Application of the Wiener–Hopf technique Higher order finite-size corrections

References Index

xv 667 668 669 673 675 685 687 705

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1 Introduction A journey of a thousand miles begins with a single step.

– Lao Tzu

This brief introductory chapter’s purpose is to direct you, the reader, quickly to those places in the book where you can find general introductory information that may be helpful for an overview of and the orientation within the book. It is deliberately kept short to avoid redundancies.

The general motivation for the writing of the book and the main targeted readerships as well as the levels of sophistication assumed and detail aimed at in the presentation of the various parts of the book are outlined in the preface. There, we also attempt a delineation of an assessment of the relevance of the book’s topics for current and potential future research.

The list of contents, by its nature in the form of key words and key phrases, provides a more comprehensive orientation of all the topics treated and their mutual dependence.

The book is divided into six major parts and, including this short introductory chapter, into twenty chapters. Each part and each chapter begins with a description detailing their respective subject matter. The part descriptions supply the bigger picture, while the outlines at the beginnings of each chapter point more specifically to the topics treated.

Wherever this seemed to be helpful, we have attempted to supply further signposts about what we have achieved and where we plan to go from there.

At various places, we also remark on the depth with which the topics are treated, what may have been left out or will only be mentioned in passing, the relation to other parts and chapters and the book’s intentions as a whole, and where to find alternative and further specialized treatments of these topics.

In order to supply a rough overview, let us briefly summarize the major parts of the book and their interrelationship.

Part I ranges from the fundamental concepts and tools required for an understanding of strongly interacting quantum matter to the fundamental models that represent the physical systems of strongly interacting quantum matter. In this book, we want to investigate selected aspects of these models with a particular emphasis on the uses of the exact methodology of the Bethe ansatz and of quantum integrability.

Part II is devoted to the quantum inverse scattering method and the algebraic Bethe ansatz that demonstrate the quantum integrability of certain one-dimensional strongly interacting quantum models and provide their exact solution. Our approach makes decisive use of the intimate connection between these models and two-dimensional

Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz. Hans-Peter Eckle, Oxford University Press (2019). © Hans-Peter Eckle. DOI: 10.1093/oso/9780199678839.001.0001

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Introduction

models of classical statistical mechanics. The concrete models we shall be enlisting in this part are the Heisenberg quantum spin chain as one-dimensional quantum model and the six-vertex model as two-dimensional classical statistical model.

In part III, we introduce the coordinate Bethe ansatz, the original approach Bethe used to solve the Heisenberg quantum spin chain. Since this approach does not allow us to understand why the models are quantum integrable, we shall address it only after we discuss the algebraic Bethe ansatz and the quantum inverse scattering method. The coordinate Bethe ansatz approach is, however, still extremely useful. We demonstrate this again for the Heisenberg quantum spin chain and also for a gas of Bosons interacting via δ-function potentials in one dimension.

Part IV is concerned with strongly interacting quantum models where the fundamental constituents have internal degrees of freedom. Our examples, the one-dimensional gas of Fermions interacting via δ-function potentials, the one-dimensional Hubbard model, and the Kondo model of a magnetic impurity interacting with conduction electrons, are all electronic models where there is only one internal degree of freedom in addition to the particle degree of freedom, which is electronic spin. We shall find that these models can be solved by two interconnected Bethe ansätze. The method is thus called nested Bethe ansatz.

Thus far, the Bethe ansatz methods discussed were mainly investigating the low-lying and therefore zero temperature properties of the quantum models. In part V we examine how to extend the Bethe ansatz to finite temperatures. Again, our quantum models of choice will be the Heisenberg quantum spin chain and also the Bose gas interacting via δ-function potentials in one dimension.

In part VI, the final part of the book, we relinquish another assumption we made or had to make in order to find solutions of the Bethe ansatz equations, equations which generally hold for a finite system. So far, we usually considered the thermodynamic limit, the limit of an infinite system size. This limit allowed us to rewrite the Bethe ansatz equations as linear integral equations for certain densities whose solutions characterized solutions of the Bethe ansatz equations, but only, of course, for the thermodynamic limit.

The Bethe ansatz for finite systems attempts to find corrections to the Bethe ansatz solutions and physical quantities, e.g. the ground state energy of the thermodynamic limit that takes into account the finiteness of a system. Again, we inquire into how this can be achieved using the Heisenberg quantum spin chain as our exemplary model system.

The focus of this book is on selected concepts, methods, and mathematical techniques in the area of strongly interacting quantum matter systems, especially the various Bethe ansatz techniques discussed. We hope that these techniques will prove useful in future research in the area of strongly interacting quantum matter. We also hope that some physical insight will be gained from the models of quantum matter used as examples to demonstrate the concepts and techniques and will provide guidance for the understanding of other systems not treated here.

For the most part, we shall use natural units where the speed of light, Boltzmann’s, and Planck’s constants are c = k ≡ kB = h̄ = 1,

(1.1)

except when including the constants explicitly will render the results more transparent.

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Part 1 Methods and Models in the Theory of Quantum Matter Cannot we be content with experiment alone? No, that is impossible; that would be a complete misunderstanding of the true character of science. The man of science must work with method. Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.

– Henri Poincaré (1854–1912)

This first part of the book presents an overview of the most important methods indispensable for an understanding of the theory of strongly interacting quantum matter.

Moreover, we introduce a selection of quantum mechanical many-particle models and the related concepts that form the background of the theory of quantum matter, especially in view of the quantum integrable models, whose exact Bethe ansatz solutions are discussed in later parts of the book. These methods and models are relevant also in many other parts of theoretical and mathematical physics. It is therefore recommended that readers review this material to judge how familiar they are with it.

These chapters, however, do not treat and do not attempt to treat their topics in a fully comprehensive manner. There is always a lot more that could be covered. In fact, there is a vast literature specifically devoted to these topics. Nevertheless, we attempt as clear and comprehensible a treatment as possible of the aspects we cover with the intention to render those aspects that we do cover self-contained. Where a self-contained treatment is beyond the limitations of this book, we provide appropriate hints to the literature specially devoted to these topics.

More specifically, in chapter 2, basic facts are reviewed from the quantum mechanics of many-particle systems, in particular leading from the Hilbert spaces representing quantum many-particle systems to a discussion of second quantization, which is the language most useful to formulate the models of strongly interacting quantum matter.

Moreover, in chapter 3 we address the quantum mechanical theory of angular momentum, especially for many quantum particles, which is indispensable for an understanding of the magnetic properties of the models of strongly interacting quantum

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4

Methods and Models in the Theory of Quantum Matter

matter. These magnetic properties will be at the centre of much of our discussion of quantum integrable models and their exact Bethe ansatz solutions.

Quantum many-particle theory is, of course, resting on the foundations of equilibrium statistical mechanics, especially quantum statistical mechanics. But classical statistical mechanics also will be necessary to appreciate the developments of quantum models that are integrable by the Bethe ansatz method. An examination of the methods and results of equilibrium statistical mechanics, both classical and quantum, will therefore be a useful addition in this first part of the book, and which we take up in chapter 4.

Among the most fascinating phenomena of many-particle systems, again classical and quantum, phase transitions and critical phenomena occupy a prominent place.

Their theoretical description is challenging and requires an arsenal of sophisticated and innovative methods that are outlined in chapter 5, where we also analyse the approach to the thermodynamic limit of systems of finite size.

There is an intimate connection between quantum field theory and (classical) statistical mechanics on which much of the quantum inverse scattering method and the algebraic Bethe ansatz is founded. Chapter 6 offers an introduction to this immensely useful connection, which will also play a central role in the subsequent chapter.

Bethe ansatz calculations for finite systems, being rather more involved than those in the thermodynamic limit, produce results that can be directly compared to predictions based on the conformal symmetry of two-dimensional classical statistical mechanics.

In order to fully appreciate this connection, chapter 7 offers an introduction into basic aspects of the conformal symmetry of critical systems.

While the chapters described so far were mainly concerned with methods useful for a thorough appreciation of the Bethe ansatz methodologies examined in later parts of the book, chapter 8 of this first part introduces the physical background of a selection of models of strongly interacting quantum matter together with methods to investigate and understand them. The selection criteria have been whether appropriate versions of the models exhibit quantum integrability and are solvable by Bethe ansatz. The quantum many-particle models considered range from the Bose fluid to models of itinerant as well as localized magnetism and to the Fermi liquid and ultimately to models of strong light–matter interaction.

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2 Quantum Many-Particle Systems and Second Quantization One of the principal objects of theoretical research is to find the point of view from which the subject appears in the greatest simplicity.

Josiah Willard Gibbs (1839–1903)

This chapter reviews some aspects of the quantum mechanics of systems composed of many particles (many-body or many-particle systems), which will prove useful for the later developments in this book. We mainly concentrate on the foundations of quantum many-particle physics leading to the formalism of second quantization as a convenient language for the formulation of the properties of the many-particle systems of quantum matter.

Many-particle quantum systems can be described by a many-particle Schrödinger equation, whose corresponding wave function depends on the configuration of the particles, e.g. their positions ri and possibly further quantum numbers, e.g. the spin quantum numbers σ i of the particles. In practice this approach is very cumbersome even for quite modest numbers of particles, let alone for the macroscopic numbers of particles of quantum statistical mechanics and condensed matter physics. Second quantization is a formulation or language of many-particle quantum mechanics that helps to minimize the technical complications of practical calculations for many-particle systems.

It is also the appropriate language of other branches of theoretical physics, most notably quantum field theory (see, for example Lancaster and Blundell, 2014).

Two examples illustrate the usefulness of second quantization. In quantum field theory as well as many applications of many-particle physics and condensed matter physics, the number of particles is variable, i.e. particles can be created as well as destroyed.1 Moreover, the Schrödinger equation, and hence the wave function, for a

1 As we shall see, the concept of creating and annihilating particles is a rather abstract one, especially for Fermions, i.e. particles for which the Pauli exclusion principle applies. Richard Feynman, in his Nobel prize acceptance speech (Feynman, 1965), alludes to this in a personal reminiscence: ‘I remember that when someone had started to teach me about creation and annihilation operators, that this operator creates an electron, I said, “how do you create an electron? It disagrees with the conservation of charge”, and in that way, I blocked my mind from learning a very practical scheme of calculation.’

Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz. Hans-Peter Eckle, Oxford University Press (2019). © Hans-Peter Eckle. DOI: 10.1093/oso/9780199678839.001.0001

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6

Quantum Many-Particle Systems and Second Quantization

system of N = 1024 particles is different from those of a system of N = 1024 − 1 particles.

Yet, we expect both systems to exhibit the same macroscopic physics. In the language of second quantization, which is specifically adapted to accommodate variable numbers of particles, we can cope easily with both situations as detailed work in this chapter and in chapter 4 on equilibrium statistical mechanics shows, as well as throughout many other chapters of this book.

Second quantization is a standard topic of quantum many-particle theory and treatments can be found in many books wholly or partly devoted to this topic. A classic reference devoted to the method of second quantization is Berezin (1966). Some standard references are the corresponding chapters of Abrikosov et al. (1975), Fetter and Walecka (2003), Mahan (2000), and Negele and Orland (1998). More recent work includes Nazarov and Danon (2013), Altland and Simons (2010), and Coleman (2015).

Of course, all of these works treat many more topics in the theory of quantum manyparticle physics.

In particular, applications of the method of second quantization can be found in the cited works and in later chapters of this book, where we shall make ample use of the language of second quantization. In this chapter, however, we concentrate on the formalism, demonstrating its power with only a few elementary examples.

Section 2.1 of this chapter constructs the Hilbert spaces, the tensor product spaces, and the Fock spaces, appropriate for the states of the many-particle system and then section 2.2, selects from these Hilbert spaces the symmetrized many-particle states of the many-particle Hilbert space that describe Bosons, where any number of particles can occupy the same quantum state and the antisymmetrized many-particle states that describe Fermions, where at most one particle can occupy the same quantum state.

This construction of Hilbert spaces is more general: for any quantum system composed of subsystems, a Hilbert space can be constructed in the way we describe.

Therefore, we initially keep the discussion more general before we focus again on Hilbert spaces composed of (many) quantum particles.

For the following few sections, we focus on the Bosonic case, before eventually also discussing Fermions. In section 2.3, we introduce creation and annihilation operators for Bosons. These operators are the main objects in which the formalism of second quantization is expressed.

The creation and annihilation operators can be expressed in different orthonormal and complete bases. The transformations of the creation and annihilation operators between different abstract orthonormal and complete bases are derived in section 2.4, while in section 2.5 the creation and annihilation operators in the position basis, then called quantum field operators, are introduced as one of the most important examples.

Section 2.6 is devoted to the introduction of one-particle operators, section 2.7 to two-particle operators in the formalism of second quantization.

An elementary introduction of second quantization starts from the time-dependent single–particle Schrödinger equation of basic quantum mechanics. How this can be achieved is demonstrated for Bosons in section 2.8 and for Fermions in section 2.10.

Section 2.9 finally returns to Fermions, introducing creation and annihilation operators for the Fermionic case.

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The penultimate section of this chapter, 2.11 demonstrates explicitly the equivalence of the many-particle wave function and the formalism of second quantization, while the final section 2.12 of this chapter touches on the issue of the correct ordering of creation and annihilation operators, i.e. the normal ordering.

As a prelude to the formalism, we start with an elementary exercise: the quantum treatment of the single particle one-dimensional harmonic oscillator in terms of creation and annihilation operators, sometimes also called ladder operators. In this exercise we are reminded of important notions that help illuminate the more formal treatment of the rest of this chapter. Moreover, the algebraic structure encountered here will reappear many times in this and in later chapters. Hence, it may be quite a good idea to go through this elementary exercise to gain confidence for the more involved later developments. .....................................................................................................................................

EXERCISE 2.1 Quantum mechanical harmonic oscillator The Hamiltonian of the one-dimensional harmonic oscillator of a mass m and frequency ω is given in terms d of the position operator x and the momentum operator p = −i dx satisfying the canonical commutation relation

[x, p] = i

(2.1)

1 2 mω2 2 p + x .

2m 2

(2.2)

as H=

Among the many ways to solve the Schrödinger equation of the harmonic oscillator, a particularly elegant, and fruitful, way deconstructs the Hamiltonian (2.2) into operators, the creation and annihilation operator, respectively

mω 1 mω 1 † a = x−i p and a = x+i p .

(2.3) 2 mω 2 mω

• Show, using the ladder operators a and a† , that the canonical commutation relation [x, p] = i becomes

[a, a† ] = 1

(2.4)

1 H = ω a† a + .

2

(2.5)

and the Hamiltonian (2.2)

• Furthermore, show that, if λ is the† eigenvalue corresponding to the normalized eigenstate |λ of the operator = a a, then

a|λ = cλ |λ − 1

(2.6)

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a† |λ = dλ |λ + 1 .

(2.7)

• Calculate the coefficients cλ and d λ .

• Prove that λ ≥ 0 and that λ = 0 must be an eigenvalue. What is, hence, the spectrum of eigenvalues of and H, respectively? Show that this implies for the ground state a|0 = 0.

• Use the representation (2.3) of the ladder operators as differential operators to solve the Schrödinger equation for the ground state corresponding to λ = 0, i.e. determine the ground state wave function x|0 .

• Determine† the wave function of the first excited state by applying the creation operator a once to the ground state wave function.

The results obtained thus far suggest that the operator can be interpreted as an operator counting the number of excitations of the harmonic oscillator. In order to make this even more suggestive let us change the notation and replace by n and |λ by |n . Furthermore, as we shall see in more detail in this chapter, these results suggest an interpretation of the excitations of the harmonic oscillator as particles or quasiparticles. The state with no particles |0 then corresponds to the vacuum state.

Show with the help of (2.6) and (2.7) that the properly normalized state of n • excitations or n particles is 1 † n a |n = √ |0 .

(2.8) n!

• Finally, to appreciate how fruitful this algebraic treatment of the harmonic oscillator is, calculate the expectation values of the first few powers of the position operator in the state |n , let us say x, x2 , x3 , and x4 . † • Hint: Prove † first1 that the number operator n = a a, and hence the Hamiltonian H = ω a a + 2 , are Hermitian operators and that therefore the corresponding eigenstates, which are non–degenerate (why?), are orthogonal. .....................................................................................................................................

With this exercise at the back of our minds, we can now start to develop the formalism of second quantization by first constructing a Hilbert space appropriate for a quantum many-particle system.

2.1

Many-particle Hilbert spaces

The formalism of ‘second quantization’2 provides an elegant and economic way to describe a physical system containing a great, possibly indeterminate, number of 2 The name is a trifle unfortunate and originates from the interpretation of the algebra of ladder operators (see exercise 2.1). These operators and their corresponding quantum excitations can be viewed as discrete ‘quantized’ units. It must be emphasized, however, that ‘second’ quantization is a representation of quantum mechanics particularly suitable for problems involving many particles. It is not a quantizing an already quantized theory. However, to be able to distinguish representations, we shall in places also have to use the equally unfortunate epithet ‘first’ quantization.

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particles. It describes particles as quanta of a quantum field and is, hence, at the heart of the modern understanding of quantum mechanics and quantum field theory.

We assume that the solution of the quantum problem for one particle is known, i.e. we assume that

• there is a one-particle Hilbert space (quantum state space) H1 , with, especially, the scalar product φ|ψ of states |ψ , |φ ∈ H1 from this Hilbert space; • this means in particular, that the one-particle Schrödinger eigenvalue problem has been solved for the single particle Hamiltonian H: H|λ = λ |λ ,

(2.9)

where |λ ∈ H1 is a normalized eigenstate and λ the corresponding eigenvalue; • furthermore, the time evolution of the particle is determined by the unitary operator (ignoring the possibility of an explicitly time-dependent Hamiltonian) U (t) = e−i Ht ;

(2.10)

• lastly, observables such as position r, momentum p, angular momentum L, etc., of the single particle problem have been determined.

Second quantization is a formalism permitting to construct quantities that correspond to a system composed of an arbitrary, indeterminate number of such quantum particles under the assumption that the statements above for a single particle hold true.

The basis of the formalism consists in constructing Hilbert spaces and states for an arbitrary number of particles from the Hilbert space and states of a single particle.

2.1.1

Composite Hilbert space of two systems A and B

As mentioned in the introduction to this chapter, we begin with a more general point of view. Assume that there are two quantum systems A and B which may but need not be individual quantum particles. For instance, system A could denote a microscopic system, while system B could represent a macroscopic measurement apparatus. Their respective Hilbert spaces are H A and H B . We are interested in the composite quantum system AB. We can construct a Hilbert space for the composite quantum system in two different ways. Both ways begin by forming a space of all ordered pairs of states taken from the Hilbert spaces H A and H B M ≡ H A × H B = F|F = ( f A , f B ), f A ∈ H A , f B ∈ H B

(2.11)

which can be made into a composite Hilbert space by choosing a scalar product in two different ways.

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2.1.1.1

Tensor product Hilbert space

The first construction is achieved through the introduction of a tensor product Hilbert space H AB of two Hilbert spaces H A and H B , whose dimensions dim H A and dim H B need not be the same, H AB = H A ⊗ H B

(2.12)

with dimension dim H AB = (dim H A )(dim H B ). With respect to the composite Hilbert space H AB , the Hilbert spaces H A and H B are called factor spaces.

It should be emphasized that already the Hilbert space of a single particle can be a tensor product space as the example of the spin-orbit Hilbert space of a single particle shows where Hspin-orbit = Horbit ⊗ C 2

(2.13)

is the tensor product of the orbital Hilbert space Horbit of the particle with its twodimensional spin state space C 2 .

Now the construction of the tensor product Hilbert space H AB proceeds as follows.

For each pair of states f A ≡ |ψ A ∈ H A and f B ≡ |ϕ B ∈ H B there is a (formal) product state F for which different notations are in use F ≡ |ψ AB ≡ |ψ A ⊗ |ϕ B ≡ |ψ A |ϕ B ≡ |ψ A , ϕ B ≡ |ψ, ϕ .

(2.14)

These notations, going from left to right, emphasize less and less that the states belong to, in general, different Hilbert spaces. Hence, their use requires more and more caution and a clear understanding of their meaning in particular situations.

The composite states are linear in each of their factors separately, i.e.

|ψ A ⊗ | λ|ϕ1B + μ|ϕ2B = λ|ψ A ⊗ |ϕ1B + μ|ψ A ⊗ |ϕ2B ,

λ|ψ1A + μ|ψ2 ⊗ |ϕ B = λ|ψ1A ⊗ |ϕ B + μ|ψ2A ⊗ |ϕ B

(2.15) (2.16)

with complex numbers λ and μ.

The scalar product between composite states is formed in a space-wise manner by ψ A | ϕ B | · |ξ A |ζ B = ψ A |ξ A ϕ B |ζ B .

(2.17)

In order to obtain a composite Hilbert space H AB large enough to contain states that cannot be written as pure product states, i.e. states of the form |ψ1A ⊗ |φ1B + |ψ2A ⊗ |φ2B

(2.18)

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we need to consider the linear span (also called linear hull) of the space M . This is achieved by taking bases of the factor Hilbert spaces H A and H B , e.g. {|nA } and {|mB }, to form a basis of the composite Hilbert space H AB {|nA ⊗ |mB },

(2.19)

a so-called (tensor) product basis, in the sense that all states of the form |ψ AB =

cn,m |nA ⊗ |mB

(2.20)

n,m

with complex numbers cn, m as expansion coefficients, i.e. the linear span of M , define the composite Hilbert space H AB .

Since an orthonormal basis in H AB satisfies, using (2.17), nA , mB |pA , qB = nA |pA mB |qB = δnp δmq ,

(2.21)

that is, the basis of the composite Hilbert space is orthonormal if the bases of the factor states are orthonormal, we obtain for the scalar product of two states of the form (2.20) ψ1AB |ψ2AB =

(1)∗ (2) cnm cnm ,

(2.22)

n,m

which completes the construction of the composite Hilbert space H AB .

Using the notion of a product Hilbert space H AB , we are now in a position to introduce in a formal way an important notion at the heart of many investigations in quantum physics. It is the notion of quantum entanglement, which goes back to important publications by Einstein, Podolsky, and Rosen 1935, and especially by Schrödinger (1935). A composite state in H AB is called entangled if it cannot be represented as a product state but only as a superposition of product states, as in (2.20).

Examples of composite states that cannot be written as product states are the so-called Bell states |

1 AB ± = √

2

|0A , 0B ± |1A , 1B

1 AB = √ |0A , 1B ± |1A , 0B , | ± 2

(2.23)

(2.24)

which are states in the composite Hilbert space H AB = H2A ⊗ H2B

(2.25)

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where the factor states describe two-level systems or so-called qubits. These states are maximally entangled states. Details about maximally entangled states and entanglement measures in general can be found in Audretsch (2007).

2.1.1.2

Composite Hilbert space as a direct sum

However, a different way is also possible to construct a composite Hilbert space from the Hilbert spaces H A and H B of the two quantum systems A and B. The starting point is again the space M of ordered pairs of states given in (2.11).

However, now we define the scalar product of two states F = ( f A , f B ) ≡ |ψ AB ≡ (|ψ A , |ψ B ) and G = (g A , g B ) ≡ |ϕ AB ≡ (|ϕ A , |ϕ B ) from M as F · G = f A · g A + f B · g B ≡ ψ AB |ϕ AB ≡ ψ A |ϕ A + ψ B |ϕ B .

(2.26)

This definition of a scalar product for the space M makes it into a composite Hilbert space H˜ AB different from the tensor product Hilbert space H AB we constructed in the previous section. This composite Hilbert space is called a direct sum space and is denoted H AB = H A ⊕ H B .

(2.27)

In order to avoid possible confusion, we mention that multiplication by a complex number λ is to be understood component-wise

λF = λf A , λf B = λ|ψ AB = λ|ψ A , λ|ψ B ,

(2.28)

in contrast to the tensor product Hilbert space H AB where

λ|ψ AB = λ|ψ A |ψ B = |ψ A λ|ψ B = λ|ψ A |ψ B .

(2.29)

We encounter this important kind of space later under the name of Fock space when we discuss situations involving particle production and destruction and, hence, need a Hilbert space with a variable number of particles.

There we shall also need the notion of orthogonality for states from different Hilbert spaces that can be conveniently defined using a direct sum of these spaces. We extend the vectors f A ∈ H A and g B ∈ H B to the space H = H A ⊕ H B by defining

F0 ≡ f A , 0 ,

(2.30)

G0 ≡ 0, g B .

(2.31)

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Calculating the scalar product (2.26) of these two special vectors of H , we discover the important result F0 · G0 = f A · 0 + 0 · g B = 0,

(2.32)

namely, that states from different subspaces of a direct sum Hilbert space are orthogonal to each other.

2.1.2

Case of many distinguishable particles

The construction of a composite Hilbert space consisting of the Hilbert spaces of two subsystems, discussed in the previous section, can easily be generalized to the case of many subsystems. From now on we focus on the case of many subsystems where each subsystem is a single quantum particle. In constructing the many-particle Hilbert space, let us assume for the time being that the particles are distinguishable. Of course, in quantum physics, we encounter many situations where particles are indeed distinguishable, e.g. the electron and proton forming a hydrogen atom are distinguishable quantum particles.

The Hilbert space of a system composed of exactly N particles is given by the tensor product Hilbert space of N copies of the single-particle Hilbert space H1 with dimension dim H1 : H1⊗N ≡

N i=1

⊗H 1 ≡ H 1 ⊗ H 1 ⊗ · · · ⊗ H 1

(2.33)

N

The states of this Hilbert space are formed from linear combinations of product states of the form (cp. (2.14)) |ϕ1 , ϕ2 , . . . , ϕN = |ϕ1 |ϕ2 . . . |ϕN ,

|ϕi ∈ H1 .

(2.34)

Such a state vector represents a state of the system where the first particle is in state |ϕ 1 , the second in the state |ϕ 2 , . . ., and the Nth in the state |ϕ N . Crucially, we note that the states of different particles may coincide, i.e. that the possibility |ϕi = |ϕj

(2.35)

for particles i = j must be taken into account.

The scalar product in H1⊗N is defined by generalizing (2.17) ϕ1 , ϕ2 , . . . , ϕN |ψ1 , ψ2 , . . . , ψN = ϕ1 |ψ1 ϕ2 |ψ2 · · · , ϕN |ψN .

(2.36)